This lacks evidence.

Starting as early as 2005, politicians have claimed to have discovered discarded prayer rugs — and perhaps even Qurans or "a lot of stuff written in Arabic" — at the southern border, hinting without evidence that Islamic migrants were entering the United States through Mexico and could pose a terrorism threat.

Mr. Trump's tweet revived this long-running and still unproved rumor. He was citing an interview in The Washington Examiner, a conservative newspaper, with one anonymous female rancher who lives on the border.

"I've never seen any Middle Easterners — I've seen prayer rugs out here — but I've never seen any myself," the woman, whose face is obscured, said in a videotaped interview.

Asma Afsaruddin, a professor of Islamic studies at Indiana University, said that prayer rugs were meant to be kept clean and doubted that they would be deserted by practicing Muslims.

"Standing in a clean place is a requirement of Islamic prayer," she said. "Many of these rugs have images of the Kaaba in Mecca and other religious symbols on them. For all these reasons, they would not be just casually tossed around or carelessly discarded to be desecrated by others."

The rancher also told The Examiner that she had heard from Border Patrol agents that migration from countries other than Mexico "has really increased in the last couple years, but drastically even in the last six months."

"Chinese, Germans, Russians, a lot of Middle Easterners," she continued. "Those Czechoslovakians they caught over on our neighbor this last summer."

"There's a lot of people coming in from not just from Mexico," the rancher added. "People, the general public, just don't get the terrorist facts of that."

That claim echoes earlier ones from Mr. Trump: baselesswarnings that "unknown Middle Easterners" had infiltrated a migrant caravan and that terrorists were pouring across the southern border.

Nowhere in the White House's 25-page counterterrorism policy, released in October, was the threat of terrorists infiltrating the nation's southwest border raised. And the State Department, in a September report, said there was "no credible evidence" that terrorist groups had sent operatives to enter the United States through Mexico.

The Department of Homeland Security did not respond to requests for evidence of Mr. Trump's claims.

The Washington Examiner report, like most of its predecessors over the past decade, did not include any photographic evidence of the prayer rugs in question and largely relied on hearsay.

"Along the Mexican border there have been stories of suspicious items picked up by local residents, including Muslim prayer rugs and notebooks written in both Arabic and Spanish," former Senator Kay Bailey Hutchison, Republican of Texas, warned in a March 2005 speech.

Later that year, former Representative Tom Tancredo, Republican of Colorado, told CBS that, during a visit to the border in Arizona, "we've found copies of the Quran, we have found prayer rugs, we have found a lot of stuff written in Arabic, so it's not just people from Mexico coming across that border."

In 2015, the right-wing outlet Judicial Watch reported, based on unnamed sources, that Muslim prayer rugs were recovered at an Islamic State training camp eight miles away from the United States-Mexico border. Federal and local officials in the United States and Mexico denied those claims.

These persistent rumors promote the message that Muslims inherently "constitute a menace and danger to our safety," Professor Afsaruddin said. "Remember Sinbad and his flying rug in the Arabian Nights? Maybe that is what they are remembering — after all, the Arabian Nights 'prove' that this is how Arabs and Muslims get around — talk about enduring stereotypes from the world of fantasy!"

The most recent news reports about prayer rugs on the southern border surfaced last summer — describing a scene in the fictional action film "Sicario: Day of the Soldado."